The functional form of the dual mixed volume (Q2070085)

The authors generalize the classical notion of dual mixed volume of star bodies to that of dual quasi-concave functions, which are defined as follows: \(f:\mathbb{R}^n\longrightarrow\mathbb{R}\) is called dual quasi-concave if \(f\bigl(\lambda x+(1-\lambda)y\bigr)\geq\min\bigl\{f(x),f(y)\bigr\}\) for \(\lambda\in(0,1)\) and \(x,y\in\mathbb{R}^n\) such that \(\langle x,y\rangle=\|x\|\,\|y\|\). Then, given \(n\) nonnegative upper semicontinuous dual quasi-concave functions \(f_i:\mathbb{R}^n\longrightarrow[0,M_i]\), \(i=1,\dots,n\), the dual mixed volume of \(f_1,\dots,f_n\) is defined as \[ \tilde{V}(f_1,\dots,f_n):=(M_1\cdots M_n)^{1/n}\int_0^1\tilde{V}\left(X_a\Bigl(\frac{f_1}{M_1}\Bigr),\dots,X_a\Bigl(\frac{f_n}{M_n}\Bigr)\right)da, \] where the functional \(\tilde{V}(\cdot,\dots,\cdot)\) in the integral is the classical dual mixed volume of the star bodies given by the superlevel sets of the functions \(f_i/M_i, i.e., \) \(X_a(f)=\{x\in\mathbb{R}^n:f(x)\geq a\}\). The main properties of this new functional are studied (monotonicity, linearity, continuity, polynomial expansion for the radial convolution of the functions...), and several classical inequalities are extended to this new setting, as the dual mixed Brunn-Minkowski inequality or the dual Aleksandrov Fenchel inequality.